Applying compactness theorem in First Order Logic

Hey!

Sorry for posting again so quickly, but I also have an issue for a second concept. This one is regarding applying the compactness theorem in first order logic, the framework is a Hilbert system:

L is a FOL (First order language) with R, where R is a single binary predicate symbol. Suppse A = ⟨V,E⟩ is a structure for this language domain V = |A|. Suppose also that E = RA, is the interpretation of the symbol R in A.

So ⟨V, E⟩ can be viewed as a directed graph; i.e., a (possibly infinite) set of vertices in V connected by edges in E.

Note that A Hamiltonian cycle in a graph is a finite sequence of vertices a1, a2,. . . , an such that the following 3 conditions are met:

How do you describe a sentence σn in the language L that has the property ⟨V,E⟩ |= σn if and only if ⟨V,E⟩ has a Hamiltonian cycle with n vertices. The question requires to give σn explicitly in the case that n = 4.

Could you provide a hint or suggestion as to how I can begin to go about this!

Many thanks!

Last edited by batchej; Mar 19th 2013 at 05:18 PM.
Reason: More detail